Fracturing at contact surfaces subjected to normal and
tangential loads
Ketan R. Shah
1
and Teng-fong Wong
Department of Earth and Space Sciences State University of New York at Stony
Brook Stony Brook, New York 11794-2100, USA.
A variety of rock engineering problems including drilling, cutting,
abrasion and milling involve rock-tool contact and indentation. The
pattern of indentation fractures and the role of slip conditions, surface roughness, tool radius, and initial aw size for an arbitrarily
loaded contact are not fully known. The present paper aims to
identify the elastic stress eld for a contact subjected to both normal and tangential loads and evaluate the condition for the fracture initiation and propagation. Stress elds within two spheres at
contact are available when either only normal load is applied or
when tangential load causes full slip conditions. It is shown here
that through appropriate superposition of the above two solutions,
stress eld under partial slip condition as well as during the unloading of tangential force may be determined. Maximum tensile stress
increases signicantly under partial-slip conditions as compared to
the full-slip case even though the same magnitude of tangential
force is applied. The location of maximum tensile stress moves inward from the trailing edge as the tangential force is unloaded. The
stress-intensity factors for a penny-shaped crack which initiates at
the contact periphery and follows the minimum principal stress trajectory are obtained and utilized to study indentation fracturing.
The dependency of critical loads on initial aw size, indenter radius, and slip-conditions is quantied. The predictions of fracture
density and spacing under a sliding indenter are achieved through
a simple estimate of shielding interaction between adjacent fractures. Relation of these evenly-spaced fractures with the formation
of wear grooves on sliding surfaces is discussed.
1
Present address: Cornell Theory Center, Cornell University, Ithaca, New York.
Preprint submitted to Elsevier Preprint
8 November 1996
1 INTRODUCTION
A wide range of shaping processes of brittle materials including cutting, drilling,
abrasion, and grinding involve indentation at dierent scales [1]. A hard indenter, pressed against a material surface, is applied both normal and tangential
forces. As a rst deviation from elasticity, existing initial aws within a brittle
material develop into fractures at certain critical loads. Hertz [2] observed the
formation of ring and cone cracks in a glass specimen while pressing it with
a hard sphere. A ring crack initiates at the contact periphery and propagates
away from the contact forming a cone. If a spherical indenter is made to slide
over the specimen surface, partial cone cracks initiate from the trailing edge of
contact and form at regular intervals of distance [3]. The initiation and propagation of these Hertzian fractures have been well-explained through elastic
stress eld under the contact and linear fracture mechanics with constant
fracture toughness [4,5].
Such Hertzian fracture processes are operative in rocks in scales ranging from
microscopic to macroscopic. During laboratory indentation of hard rock, fractures initiating at the contact edges move away from the contact and often
merge with the free surface forming a chip (e.g. [6]). The formation of Hertzian
fractures is followed by intense crushing and microcracking in the interior cone.
The process of chipping is sensitively dependent upon the interaction of the
inelastic core, the fractures, and the indenter [7]. The understanding of these
interaction eects is pivotal in developing better and more energy ecient
rock fragmentation techniques.
A global stress eld when applied to a porous medium induces contact normal and tangential forces on the grain scale (e.g. [8]). As the global stresses
are increased, contact forces also increase and may lead to the fracturing of
grains into smaller pieces or grain crushing [9,10] and to the overall reduction
of grain size, dened as comminution. Microstructural observations have identied Hertzian fracture at the contact of impinging grains to be the dominant
comminution mechanism in clastic rocks [11]. The crushing reduces the porosity and mechanical stiness and also blocks the uid-ow paths decreasing
the permeability [12,13]. Grain crushing also plays an important role in the
behavior of granular media at elevated pressures such as for deep geological
sediments and fault gouge zones [14,15] as well as has several engineering consequences such as for the design of ecient comminution process and oil-well
stimulation technique of perforation [16].
A sliding rough surface is covered with asperities which act as indenters and
slide relative to the other surface under full-slip conditions. The stress intensity
in the vicinity of the sliding asperities may be suciently high for the development of a trail of Hertzian chatter cracks (wear grooves). Microscopic wear
2
grooves in laboratory sliding surfaces [17,18] have geometric attributes similar
to macroscopic striations commonly observed in fault zones and glacial scour
terrains [19]. It has been suggested that wear groove characteristics (depth
and spacing) may provide useful information about the sliding history and
paleoseismicity.
The boundary conditions and loading paths of interest in these rock mechanics problems can be very complex. The (microscopic or macroscopic) contacts
may either be static (under partial-slip conditions) or sliding, and they may be
subjected simultaneously to both normal and tangential loads. Many aspects
of the fracture mechanics are not fully understood. In particular, the patterns
of fracturing under partial-slip conditions and dierent loading paths are not
well established [20]. The purpose of this paper is to model the indentation
fracturing under both full- and partial-sliding conditions. We formulate a superposition technique by which the elastic contact stress eld induced by a
spherical indenter can be analytically determined. This allows us to perform
a fairly complete analysis of the fracture initiation and propagation. The inuences of initial aw-size, indenter radius, material parameters, and loading
paths on the Hertzian fracture process are explored, and the results are applied to pertinent rock mechanics problems of wear at sliding surfaces. The
results for fracturing at grain contacts based on the present analysis will be
published elsewhere [21].
2 INDENTATION STRESS FIELD
2.1 Normal loading
Hertz [22] established the contact area and traction distribution between two
elastic spheres subjected to normal compression load and also provided the
stress elds. Assuming dimensions of the contact area to be much smaller than
that of the bodies at contact and considering each body as an elastic half-space
for the purpose of relating local deformations with contact pressure, radius of
the circular contact area a is
a=
3P R 1=3
4E (1)
where P is the applied normal load and R and E are given by
1= 1 + 1
R R
R
1
3
2
(2)
1 = 1 ? 12 + 1 ? 22
E
E1
(3)
E2
and R2 are the radii of two spheres and E1 and 1 are the elastic modulus
and Poisson's ratio, respectively, of the rst sphere. E2 and 2 are similarly
given for the second sphere. For the case of spherical indentation of a halfspace, R2 is innity and R becomes the same as the indenter radius R1 . The
variation of contact pressure, p, is axisymmetric:
R1
p(r) = po
n
1 ? (r=a)2
o1=2
where r is the distance from the center of contact and
contact pressure at the center given by
3P
po =
2a2
(4)
po
is the maximum
(5)
The stresses in either spherical body may be obtained from the solution of
an elastic half-space subjected to above normal pressure distribution. These
stresses are accurate only in the proximity of the contact since the eects of
shape and niteness of the body are disregarded. The stress eld for normal
load P and contact radius a is denoted as
ij
= ijN (P; a)
(6)
where ij is a stress tensor and ijN is the stress eld under normal contact
rst given by Huber [23]. In this paper, tensile stresses will be considered to
be positive, and the components of the above stress tensor are taken from
Hamilton [24]. The maximum tensile stress which is radial and located at the
edge of the contact circle at r = a (Fig. 1(a)), is believed to be responsible for
the formation of ring and cone cracks. The maximum principal stress contours
on a vertical plane are shown in Fig. 1(a); the tensile stress drops to zero at
the depth of 0.05a under the contact edge. The von Mises shear stress dened
as
q
h
2 + 2 + 2
J2 = xy
xz
yz
o1=2
n
1
2
2
2
(7)
+ 6 (xx ? yy ) + (xx ? zz ) + (zz ? yy )
is also contoured on the vertical plane (Fig. 1(b)). The maximum value is
0:374po for 1 = 2 = 0:25 and it occurs at the depth of 0:5a under the center
of contact.
4
P
2a
0.01
0.005
0.01
0.005
Compression
Zone
0.0
Crack
Trajectory
0.005
(a)
-2
-1
0.
09
0
0
0.17
0.25
0.33
1
2
0.09
0.37po
1
2
(b)
Fig. 1. Contours of maximum principal stress (a) and of von Mises shear stress (b)
in a half-space during normal indentation. Poisson's ratio is assumed to be 0.25.
Three minimum principal stress trajectories initiating at x = ?0:8a, x = ?a, and
x = 1:2a, along which the penny-shaped fractures are assumed to grow, are also
shown.
If the materials at contact are elastically dissimilar, self-equilibrating shear
tractions arise at the contact [25]. The shear tractions reduce the maximum
tensile stress and increase fracture load for less compliant indenter [26]. They
may be insignicant compared to the tractions generated due to sliding force
unless the mismatch in elastic constants is large [20]. Eects of these shear
tractions on stress distribution will be disregarded, but the evaluation of contact area will be done based on actual dissimilar elastic parameters (equations
(1) and (3)). In subsequent discussions, the parameters E and will be used
to denote elastic modulus and Poisson's ratio of the material in which failure
is of interest.
2.2 Tangential loading: shear traction and loading path
For elastically similar bodies, shear tractions are present only when a tangential load is applied. Cattaneo [27] and Mindlin [28] independently obtained
the distribution of shear tractions on the contact surface when a tangential
force is also applied. They assumed the Poisson's ratio to be zero for both the
5
P
Q
2s 1
R1
Unloading
2s 2
2s
Partial-slip
Full-slip
R2
2a
Q
P
Fig. 2. Distribution of contact shear tractions for dierent loading conditions from
Mindlin and Deresiewicz [30]. For full-slip case, the distribution is similar to the
normal Hertz pressure. For partial-slip conditions, a slip annulus surrounds a central
stick zone of radius s, whereas during unloading a reverse-slip annulus initiating at
contact periphery surrounds the distribution similar to partial-slip case.
spheres and that shear tractions at the contact act in the same direction as the
tangential force. The eects of Poisson's ratio may be negligible if the spheres
are elastically similar [29]. The contact area and normal pressure distribution
are determined by normal load alone and are given by equations (1) and (4).
Additional complexity is introduced by the tangential loading in that the
solution is now path-dependent. The rst results were for shear traction distributions corresponding to a loading path with P applied rst and then Q
increased [28]. As the tangential load is quasi-statically increased, shear tractions overcome friction near the contact edges and a slip annulus grows (Fig.
2). Shear tractions in a remaining region around the center are less than the
friction coecient f times normal pressure p and the region is denoted as stick
zone. When tangential load Q is equal to f P , stick zone reduces to a point
and two bodies slide freely over each other. The shear traction distribution q
is a function of radial distance r only and is given by [28]:
q (r) = f po
q (r) = f po
o1=2
n
1 ? (r=a)2
n
o1=2
1 ? (r=a)2
sra
?
s n
a
1 ? (r=s)2
o1=2 (8)
r<s
(9)
where s is the radius of stick zone and is obtained by equilibrating the above
shear tractions with the applied load Q:
s=a
1?
6
Q
fP
!1=3
(10)
Under the full-slip conditions of Q = f P , the stick-zone radius becomes zero
and distribution of q (equation (8)) is similar to that of normal pressure (equation (4)).
Mindlin and Deresiewicz [30] later extended the analysis to loadings involving
an oblique force and to loading-unloading cycle. If both P and Q are simultaneously applied (oblique force) such that Q=P < f , then slip does not take
place anywhere within the contact circle and shear traction distribution is
q (r) =
o1=2
Q n
po 1 ? (r=a)2
P
(11)
It can be seen that the distribution is the same as the full-slip case (equation
(8)) with Q=P substituting the friction coecient f .
The unloading case is when the tangential force Q is reduced from its peak
value Q which was applied after the normal load P . A zone of reverse slip emanates from the contact periphery and moves radially inward until it matches
with original slip annulus at Q = ?Q . The traction distribution for Q = ?Q
is the same as Q = Q with the sign reversed, but is rather complex for intermediate stages (Fig. 2).
q (r) = ?f po
q (r) = ?f po
n
o1=2
n
2 o1=2
1 ? (r=a)2
?2f po
q (r) = ?f po
?f po
1 ? (r=a)
s n
2
n a
1 ? (r=s2)2
1 ? (r=a)2
s n
1
a
o1=2
ra
s2
o1=2
?2
s n
2
o1=2
1 ? (s1 =a)2
a
s1
(12)
r s2
1 ? (r=s2)2
o1=2 r s1
(13)
(14)
where s1 is the radius of original stick-zone and is given by equation (10) with
Q substituted by Q and s2 is the inner radius of the annulus of reverse-slip:
s2
=a 1?
Q ? Q
2f P
!1=3
(15)
2.3 Elastic stress eld: analytic results obtained by superposition
The stress eld below the tangentially loaded contact surface was not known
until Hamilton and Goodman [31] provided the complete solution for the case
of full-slip or sliding contact. It requires the ratio of tangential to normal forces
7
to be equal to the coecient of friction of the contact surface, which in turn
implies that the shear traction distribution is obtained by just multiplying the
Hertz pressure distribution (equation (4)) with the friction coecient.
3Q n1 ? (r=a)2 o1=2
q (r) =
(16)
2a2
where 3Q=2a2 is the same as f po. The stress eld for a tangential load Q
distributed over a contact circle (of radius a) according to the above equation
will be denoted as ijS :
ij
= ijS (Q; a)
(17)
In a subsequent paper, Hamilton [24] provided more explicit expressions for
this stress eld, which is used in the present study.
To our knowledge, there have not been any previous attempts to generalize
these analytic results from the full- to the partial-slip case. Numerical integration was used by Chiang and Evans [20] to evaluate the stress eld. Since the
indentation fracture behavior hinges on the stress intensity factor which has
to be evaluated through a second numerical integration of the stresses, Chiang
and Evans' [20] approach requires the computation of a double integral which
may involve considerable numerical inaccuracy. In this study, we formulate an
analytic approach by which the stress elds for the partial-slip and unloading
cases can be explicitly obtained by superposition.
Our approach is based on the observation that the traction distributions for the
partial-slip case (equations (8)-(9)) and unloading case (equation (12)-(14))
can be considered as the superposition of the full-slip shear tractions of the
form of equation (16) distributed over two or more concentric circles of contact.
Since the materials in contact are elastic and the stress eld induced by the
full-slip traction distribution over a contact circle of given radius is given by
equation (17), the latter solution may be suitably superposed to obtain stresses
for both partial-slip and unloading cases. A careful observation of Mindlin and
Deresiewicz [30] reveals that all the possible cases of contact under normal and
tangential load may be treated in a similar manner to evaluate the stresses.
The complete stress eld for full-slip condition is the superposition of Hertzian
one and that due to tangential load [24]:
ij
= ijN (P; a) + ijS (Q; a)
(18)
where Q must be equal to f P . This stress-eld remains valid if P and Q are
applied simultaneously with Q=P < f . For the partial-sliding case in which Q
8
is applied after P and Q < f P , traction distribution is given by equation (8)
and stress eld is obtained by superposing the eld due to a negative tangential
load Q ? f P (over the stick zone of radius s) to the full-slip solution:
ij
= ijN (P; a) + ijS (f P; a) + ijS (Q ? f P; s)
(19)
The case of unloading the tangential force is more complicated (equation (12)),
but the stress eld can still be obtained by appropriately superposing positive
and negative shear tractions over three concentric circles of contact:
ij = ijN (P; a) + ijS (?f P; a)
+ ijS (Q ? f P; s1) + ijS (Q ? Q + 2f P; s2)
(20)
To illustrate the inuence of tangential loading and loading path, we consider
three cases with the same normal force P and tangential force Q such that
Q=P = 0:5. The rst loading case is of full-slip for which f = 0:5. This case
is also valid for oblique loading with Q=P = 0:5 given that f is greater than
or equal to 0.5. The second case is when f = 1:0 and Q is applied after P
and so the contact is under partial-slip condition. The last case is considered
by subjecting the contact to full-slip for f = 1:0 and unloading only Q to
0:5P and is denoted as unloading case. The maximum principal stress and
von Mises shear stress are normalized with respect to the maximum contact
pressure po, and their contours are plotted on a vertical plane parallel to the
applied tangential load cutting through the center of the contact circle.
The variation of radial stress xx at the surface for all the four cases is shown
in Fig. 3. The distribution is symmetric around the center of the contact for
normal loading case. Higher values of radial stress for full-sliding and partialslip cases are evident. The value at the center is the same for all the cases and
is approximately ?0:8po. The unloading moves the maximum radial stress
location inward and increases the compressive stress near the leading edge.
2.3.1 Full-slip case
The stress eld for the full-slip case (Fig. 4) diers from the pure normal
loading case (Fig. 1) in several important respects. First, the stress distribution
is no longer axisymmetric. The maximum values of the tensile and von Mises
shear stresses magnitude are both attained at the trailing edge. The location
of maximum shear stress shifts to be on the surface at approximately f = 0:25.
Second, the maximum values of the stresses are signicantly higher. For the
tensile stress it is more than 6.0 times the corresponding maximum for the
normal loading case (Fig. 4(a)), and for the shear stress it is 1.5 times (Fig.
9
Radial stress/contact pressure
0.80
0.40
0.00
-0.40
-0.80
-1.20
-1.50
-1.00
-0.50
0.00
0.50
1.00
Distance from contact center/contact radius
Normal loading
Full sliding
Parial sliding
Unloading
1.50
Fig. 3. Variation of horizontal radial stress on the surface in the plane containing
the applied loads. The distribution is symmetric for normal loading case with its
maximum at the contact periphery. For partial and full-slip cases, its maximum is
at the trailing edge and it is compressive at the leading edge. Unloading shifts the
location of maximum inward as well as induce tensile stress at the leading edge.
Q
P
2a
0.2
0.05
Compression
Zone
0.00
0.01
Crack
Trajectory
(a)
0.58po
0
2
2
0.
12
0.0
2
2
0.4
0.1
2
1
1
0.52
0.32
-1
22
0.
0
-2
(b)
Fig. 4. Contours of maximum principal stress 1 and minimum principal stress 3
trajectories (a) and contours of von Mises shear stress (b) for full-slip case with
friction coecient of 0.5. Maximum tensile stress has the same value as maximum
contact pressure po . 3 trajectories near the trailing edge along which fractures grow
are also shown.
10
Q
P
2a
0.2
Compression
Zone
0.05
0.00
0.01
Crack
Trajectory
(a)
0.74po
-2
-1
0
0.41
1
2
0.29
0.29
17
0.
0
0.05
1
2
(b)
Fig. 5. Contours of maximum principal stress 1 and minimum principal stress 3
trajectories (a) and contours of von Mises shear stress (b) for partial-slip case with
friction coecient of 1.0 and the ratio of tangential to normal loads of 0.5. Fractures
initiate at the trailing edge and follow 3 trajectories.
4(b)).
2.3.2 Partial-slip case
These qualitative features of the sliding case are preserved for the partial-slip
case as seen in Fig. 5. However, the stress magnitudes are quite dierent from
that in Fig. 4. In accordance with equation (10), slip is limited to the annular
region with 0:79 r=a 1. The size of compression zone is larger, and the
maximum value of the tensile stress is enhanced by a factor of 1.33 relative
to the full-slip case. Nevertheless, the tensile stresses at a distance more than
0:25a are indistinguishable from the full-slip case. The von Mises shear stress
is maximum at the trailing edge and has a 28% higher value than the full-slip
case. In general, the partial-slip condition is eective in enhancing the tensile
and shear stresses in the near vicinity of the contact, which may promote the
initiation of failure in both brittle and ductile modes.
11
Q
Maximum
tensile stress
P
2a
0.2
0.1
Compressive
Zone
(a)
0.92p
o
-2
-1
0
0.80
1
2
0.60
0.20
0.40
0
1
2
(b)
Fig. 6. Contours of maximum principal stress 1 and minimum principal stress 3
trajectories (a) and contours of von Mises shear stress (b) for unloading case with
friction coecient f of 1.0 and the ratio of tangential to normal loads of 0.5. The
previously applied maximum tangential load corresponds to full-sliding or Q = fP .
2.3.3 Unloading case
For the unloading case shown in Fig. 6, the maximum tensile stress occurs
inside the contact near the trailing edge at around x = ?0:9a and is 27%
more than the value for the full-slip case (Fig. 4). The location coincides with
the inner boundary of the reverse slip annulus. The tensile stress may induce fracturing inside the contact circle and causes the surface damage within
the reverse-slip annulus during repeated loading-unloading cycles, as was suggested by Mindlin [32]. The maximum shear stress occurred at the same location as tensile stress and has 56% higher value than the full-slip case.
3 INDENTATION FRACTURING
The indentation of brittle materials through a hard sphere has been used
to quantify the eects of surface damage by impacts as well as to evaluate
the fracture properties of the material [33,4,34,35,20,36]. There have been
two key observations. First, Auerbach [33] found that the load required for
12
the formation of Hertzian cone cracks is proportional to the radius of the
indenter, known as Auerbach's law. Second, Roesler [4] demonstrated that
the cone crack is a stable fracture system and the length of cone crack is
proportional to the P 2=3 . This relationship is referred to as Roesler's law.
The fracture mechanics of Hertzian fracture under normal loading was thoroughly analyzed by Frank and Lawn [34]. They evaluated the stress-intensity
factor for a crack initiating at the contact edge and studied the stability behavior for the propagation of an initial aw. They quantied the eects of
initial aw size and dened the range of the radius of indenter in which Auerbach's law may be valid. When an indenter is moved across the specimen
surface, evenly spaced \partial" cone cracks form at the trailing edge of the
contact circle [3]. Lawn [5] utilized the stress elds obtained by Hamilton and
Goodman [31] to evaluate stress-intensity factors for a crack emanating from
the trailing edge of a sliding indenter. For the partial-slip case, Chiang and
Evans [20] evaluated the stress intensity factor by numerical computation of
a double integral.
A key assumption in the work of Lawn [5] is that the fracture does not alter
the contact radius a and Hertz-Mindlin formulae for contact traction distribution (equations (4), (8)-(9), (12)-(14)) remain valid. When the fracture size
is comparable to contact radius, such nonlinear interaction eects is expected
to be important and signicant deviations in the contact dimensions and the
traction distribution may occur. The analysis of Lawn [5] allows the analytic
evaluation of stress intensity factors and is useful in capturing the rst-order
eects of contact tractions on indentation fractures.
3.1 Stress intensity factor and strain energy release rate
In this study, we have circumvented the intense computation required for the
double integral by rst using superposition to derive the explicit results for
the stress eld. This allows us to map out the ne details of the stress eld and
to evaluate the fracture mechanics parameters for a relatively complex stress
eld, such as that due to unloading (Fig. 6 and equation (20)). Following
Frank and Lawn [34], we assume that a partial cone crack propagates along
the minimum principal stress, 3 , trajectory. Our calculations show that the
crack is expected to follow a curved trajectory (Figs. 1(a), 4(a), and 5(a)), in
discrepancy with Chiang and Evans' [20] assumption of a vertical trajectory.
As we will discuss later, the 3-dimensional geometry of the minimum principal
stress trajectory is also dierent from the plane-strain conguration assumed
by Lawn [5]. A better representation seems to be a penny-shaped crack which
grows along the minimum principal stress 3 trajectory.
13
We shall assume that a half penny-shaped aw initiates from the trailing
edge (Fig. 1). As shown in Figs. 1(a), 4(a), and 5(a), the trajectories which
initiate near contact periphery and move away from the contact circle are
steeper when tangential load is present. If a suitable aw is not available at
the trailing edge, fracture may propagate from a near-by location and follow
the other trajectories shown in these gures [37]. The analytical stress elds
obtained in previous section is used to nd 3 trajectory and evaluate normal
stress 1 along the trajectory. The stress intensity factor for a penny-shaped
crack of length c along the trajectory is given by (e.g. [38]):
Zc b
2
p 21
KI = p
c
0
c
(21)
? b2 db
where KI is the mode I stress-intensity factor and b is the coordinate
along
p
the trajectory. The stress intensity factor can be normalized by po a :
K
=
KI
p
po a
r a Zc=a ( =p ) b=a
2
d
= c q 1 2o
2
(
c=a
)
?
(
b=a
)
0
b
a
!
(22)
where 1 is a function of non-dimensional length b=a and hence normalized
stress-intensity factor K depend on c=a . The normalized stress-intensity factor
would also depend on loading parameters such as on f for full-slip
p and on f
and Q=P for partial-slip. The integrand in equation (22) has 1= c2 ? b2 term
which is singular at b = c and equal-weight abscissas or roots of rst-order
Chebyshev polynomial are used as the evaluation points to calculate the right
hand side of equation (22) [39].
Strictly speaking, the above integral applies only to a planar penny-shaped
crack in an innite medium. For mathematical convenience, we have mapped
the curved trajectory onto a planar surface. The assumption of axisymmetry
ensures that stress-intensity factor does not vary along the crack front. It is
justied on the ground that the normal stress is more or less the same as the
principal stress along the trajectory for up to the angles of 45 degrees with the
plane containing the trajectory and normal and tangential loads (Fig. 7). The
crack-normal stress perpendicular to the trajectory or along the surface intersection is considerably higher and implies that stress-intensity factor would
be higher for the crack front on surface [40]. The actual crack-shape may be
elliptical with its major axis lying on the surface. The axisymmetry is reasonable almost up to 60 degrees and prediction of the depth of the crack would
be more accurate than its surface trace. The intersection of the crack with
the specimen surface is presumed to be straight, but in reality, the crack may
actually be curved and follow the intermediate stress trajectory on the surface
[5].
14
Normal stress/contact pressure
1.4
Along the
trajectory
1.2
1.0
45 degrees with
the trajectory
0.8
75 degrees with
the trajectory
0.6
0.4
Perpendicular to
the trajectory
0.2
0.0
0.0
0.5
1.0
1.5
Distance along the trajectory /contact radius
2.0
Fig. 7. Variation of crack-normal stress at dierent angles with the plane containing
the applied loads and the 3 trajectory. Stresses drop rapidly with distance and are
insignicant for distances higher than 2:0a. Stresses increase with the angle and are
maximum when the direction is perpendicular to the plane.
1.E-1
KI2/po2π a
Unloading
f=1; Q/P=0.5
f=1
Q/P=0.5
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
1.E-2
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
f=Q/P=0.5
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
1.E-3
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
f=1
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
Q/P=0.1
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
1.E-4 AAAAAAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAAAAAA
AAAA
AAAA
AAAAAAAA
f=Q/P=0.1
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
AAAA
1.E-5
AAAA
AAAA
C0
1.E-6
1.E-4
1.E-3
Q=0
C1
C2
C3
1.E-2
1.E-1
1.E+0
Crack size/contact radius
1.E+1
Fig. 8. Normalized energy release rates or square of stress-intensity factors
(KI2 =p2o a) as a function of crack-length normalized by contact radius for various
cases of normal and tangential loadings. The variation has two peaks and unstable
branches Co and C2 for normal loading case. Stress intensity factors are increasing
with tangential force and would reduce the required loads for fracture initiation.
To establish contact with the Frank and Lawn's [5] previous analysis, we
consider the Hertzian fracture evolution in term of the strain energy release
rate G, dened by KI2 (1 ? 2 ) =E . Results for the square of the normalized
stress intensity fracture or normalized energy release rate GE= (1 ? 2 ) p2o a =
KI2 =p2o a for dierent full-slip, partial-slip, and unloading conditions are shown
in Fig. 8 as a function of normalized crack length c=a. In the absence of any tangential force, the variation shows two peaks similar to Frank and Lawn's [34]
calculation for plane strain crack. The rst and second peak corresponds to the
initiation and propagation of a ring and cone crack respectively. The initial aw
of size cf may grow when the condition KI = KIc (or G = KIc2 (1 ? 2 ) =E ) is
met where KIc is the fracture toughness of the material. The growth will be
15
unstable if the slope of the G versus c curve is positive (branches C0 and C2
in Fig. 8) and otherwise it will be stable.
For the full-slip case, Lawn [5] showed that G as a function of c has only
one peak unless the friction coecient is anomalously low ( f < 0:02). Our
calculations demonstrate that Lawn's [5] conclusion is generally applicable for
both partial- and full-slip conditions (Fig. 8). This implies that the formation
of partial cone crack is the preferred mode of failure in the presence of tangential loading. For a given crack length, tangential loading can signicantly
enhance the stress intensity and strain energy release rate, especially for crack
dimensions smaller than the contact radius.
For the same normal and tangential loads, KI and G are higher for the partialslip case before the peak, as can be seen by comparing the full-slip case of
f = 0:5 with partial-slip case of f = 1:0 and Q=P = 0:5 in Fig. 8. This implies
that initiation of unstable propagation of a cone crack occurs at a lower load
for the partial-slip case. However, after the peak KI and G values are higher
for the full-slip case. The unstable propagation of a cone crack for the full-slip
case will actually extend further before it can be arrested.
Variation of the square of stress intensity factor for the unloading case is also
shown in Fig. 8. The crack is assumed to initiate inside the contact circle at the
inner edge of reverse slip annulus where tensile stress is maximum (Fig. 6(a))
and propagate along minimum principal stress trajectory. The stress intensity
factors are compared with the full- and partial-slip cases with Q=P = 0:5. KI
is higher than the full-slip case and is comparable to the values for partial-slip
case prior to the peak.
3.2 Critical load
Critical load during the spherical indentation of brittle materials is dened as
the load required for a cone cracks to form [5,20]. One of the simplest criterion
is obtained by equating the maximum value of principal stress with the tensile
strength of brittle materials [35]. For full-slip or sliding case, the maximum
value occurs at the trailing edge and acts in x direction:
3
P 1 ? 2 4 + xx =
2a2 3 + 8 f = t
(23)
where t is the material tensile strength. Substituting for the contact radius
(equation (1)) in the above equation and assuming the rigid indenter of radius
16
Critical normal load (Newtons)
1.E+4
Based on a curved
penny-shaped flaw
1.E+3
Based on maximum K for
vertcal penny-shaped flaw
1.E+2
Based on a long
plane-strain flaw
1.E+1
Based on maximum K
for curved pennyshaped flaw
1.E+0
Partial Slip
with f=1.0
1.E-1
0.0
0.2
0.4
0.6
0.8
Ratio of tangential to normal load
1.0
Fig. 9. Critical normal loads for indentation fracturing based on: (i) maximum KI
for a vertical penny-shaped crack initiating at the trailing edge; (ii) a plane-strain
crack growing along curved 3 trajectory; (iii) maximum stress intensity factor; and
(iv) a penny-shaped crack propagating along curved 3 trajectory.
R
pressing on the elastic-brittle half-space gives
2
(1
?
2) 3 2
2
3
R
Pcr =
3 [(1 ? 2 ) =3 + f (4 + ) =8]3 E 2 t
(24)
where Pcr is the critical normal load. For oblique loading such that Q=P < f ,
the above expression still applies if f is replaced by Q=P . The critical load is
proportional to the square of the indenter radius R, contrary to Auerbach's
law and it decreases rapidly with increasing friction coecient f . This type
of criterion was adopted recently by Papamichos et al. [16] to model grain
crushing and shear compaction in porous rocks.
A major shortcoming of the above criterion is related to the diculty to unambiguously determine the parameter t , which is sensitively dependent on the
geometric attributes of pre-existing aws in the material. A more realistic criterion should be based on fracture mechanics, by equating the stress-intensity
factor of a typical aw to the fracture toughness of the material. The equivalent approach will be to compare energy release rate G with twice the surface
energy since G is related to the stress-intensity factor:
1?
G = KI2
E
2
= 2
(25)
Note that the surface energy and fracture toughness are related: = KIc2 (1 ?
2 )=2E .
The critical loads for full- and partial-slip cases are shown in Fig. 9. The
17
parameters were chosen to be E = 70 GPa, = 0:25, KIc = 1 MPa-m1=2 ,
R1 = 1 mm and R2 = 1. The pre-existing aw length was assumed to be 1
m. As expected, the application of tangential load reduces signicantly the
critical normal load. The eect of crack geometry is illustrated by comparing
with Lawn's [5] approach, which assumes a plane-strain conguration for the
crack path. Our prediction for a penny-shaped crack is generally higher by a
factor of 4, because the 3-dimensional geometric constraint in our model acts
to alleviate KI and therefore a higher load is required to reach the critical
stress intensity factor.
While the critical load should depend on the aw size, an estimate of the
critical load which is independent on the aw size may be obtained by if we
assume the existence of a aw with the most critical size corresponding to the
peaks in the curves in Fig. 8 [20]. To the extent that the pre-existing aws are
shorter than the critical dimension, this approach will provide a lower bound
on the critical load. If we identify from our computed curves the peak as
2 p2 a 1 ? Gmax = Kmax
o
E
2
= 2
(26)
where Kmax is the maximum normalized stress-intensity factor, then we can
substitute from equations (5) and (1) into the above to obtain the lower bound:
Pcr
= 4R
2
3Kmax
(27)
The lower bound critical load is proportional to the indenter radius similar
to Auerbach's law, and the slope of Pcr versus R can be used to estimate the
surface energy.
We have chosen for the pre-existing aw a dimension which is probably appropriate for comminution processes at the grain scale. It can be seen from Fig.
9 that for this situation the lower bound may underestimate the critical load
by a signicant margin. The inuence of geometric complexity is illustrated
by comparing our approach with that of Chiang and Evans [20] who assumed
a vertical trajectory. The discrepancy between the two estimates of the lower
bound are primarily for small tangential load, for which the Chiang and Evans
[20] estimates are higher by a factor of 4 when compared with the results of
present approach. The eects of slip-conditions and loading path on the critical load can also be seen in Fig. 9. Critical load for partial-slip with f = 1:0
is lower than the value for full-slip case for the same Q=P . The reduction is
maximum for Q=P = 0:2 and is by a factor of 5 and it decreases to approach
the full-slip value at Q=P = 1:0.
18
400
Normal load (Newtons)
350
300
250
200
150
100
0.0001
0.001
0.01
0.1
1
Crack-size (mm)
Fig. 10. Evolution of an initial aw into a fracture during normal loading and
dependency of critical load on aw-size. Two unstable growths for aw-size of 0.0007
mm is evident. Increasing or decreasing the aw-size, only one unstable growth
remains. Critical load generally increase as the aw-size is reduced.
3.3 Role of aw size and indenter radius
The stress and lower-bound criteria eectively disregard the role of pre-existing
aw size. The inuence of aw size on critical load has been discussed by Lawn
[5] and Cook and Pharr [36]. The variation of stress-intensity factor with cracklength for normal indentation with stable and unstable branches are shown in
Fig. 8. The C3 branch corresponds to the stable growth of Hertzian fracture
and we will calculate the load required for an initial aw to reach C3. Variation
in crack-size with load is shown in Fig. 10 with two unstable growths indicated
by increase in crack-size at the same load for normal indentation. If the aw
size is smaller, then load is higher and there would be only one unstable
regime (Fig. 10). The rst unstable growth disappears on increasing the awsize, but the critical load remains the same. If the aw-size is such that it lies
on branches C1 or C2 , then critical load is independent of aw-size. It also
implies the proportionality of the critical load with the indenter radius as for
the lower-bound criterion (equation (27)). For tangential loading branches, C1
and C2 are absent and Auerbach's law would not be valid. The aw size plays
a vital role in determining the critical load and will generally decrease the
indentation strength of material.
The critical load generally increase with the indenter radius [35] and is linearly
proportional to the radius within the Auerbach range. If maximum tensile
stress determines the critical load then the load is proportional to the square
of the indenter radius. The complete variation of load with the radius can be
studied based on the known normalized stress-intensity factors (Fig. 8). Initial
aw size is kept constant and the critical load versus the radius is shown on a
19
Critical load (Newtons)
1.E+8
f=0.0
1.E+6
1.E+4
f=0.3
1.E+2
f=0.5
1.E+0
1
10
Indenter radius (mm)
100
Fig. 11. Variation of critical loads with indenter radius for a given initial aw-size.
For small radii and normal indentation, the critical load is proportional to the
radius similar to Auerbach's law (shown by dotted lines). At large radii or when the
tangential force is present, the slope is higher than 1.0.
log-log plot in Fig. 11. The slope of 1 will correspond to the Auerbach's law.
The law is valid for small radius during normal indentation, but for higher
radii the slope is larger. The overall behavior is similar to the experimental
observations of Gilory and Hirst [35]. They found maximum value of the slope
to be 2.0 and concluded that at large radii the failure is governed by critical
stress.
4 DEVELOPMENT OF WEAR GROOVES UNDER A SLIDING
INDENTER
In the context of rock mechanics, there are also applications in which either the
constitutive behavior or the loading geometry are so complex that additional
features need to be incorporated into the model in order to capture the key
physical attributes. In drilling applications involving low-porosity crystalline
rocks in geothermal environments and high-porosity sedimentary rocks under
elevated pressures, the magnitudes of the mean and deviatoric stress eld in
the compactive zone underneath a spherical indenter are suciently high that
plastic yielding phenomena may occur [41,42]. In that case, the constitutive
model should be generalized to an elastoplastic rheology. There are many
geotechnical and geotectonic problems for which an elastic-brittle rheology is
applicable. However, in some instances the loading and fracturing processes
involve a multiplicity of indenting asperities. The comprehensive analysis of
such geometric complexities is beyond the scope of the present study. Instead
we will derive analytic estimates for the spacing of wear grooves on the basis of
insights gained from our detailed consideration of the stress eld and fracture
mechanics of a single spherical indenter.
It is commonly observed that a periodic array of fractures develop at the wake
20
of a sliding indenter. Keer and Kuo [40] obtained the spacing between cracks
in a brittle elastic half-space. They accounted for shielding caused by a neighboring crack and formulated a numerical scheme by which the fracture density
can be evaluated. Bower and Fleck [43] applied 2-dimensional boundary element method to obtain crack path and spacing for a sliding cylinder and found
that spacing is generally large enough to prevent generation of wear particles
through coalescence. Here we derive analytic estimates for the 3-dimensional
problem of a sliding spherical indenter, that will consider shielding mode of
interaction and predict spacing between neighboring cracks. The eects of
friction coecient and of applied loads will be explained.
4.1 Spacing of fractures
The analysis in previous sections allows us to study initiation and propagation of individual fractures under an indenter. Initiation at critical load is
generally unstable, but further propagation takes place in a stable manner.
Critical normal load is extremely sensitive to the applied tangential load and
slip conditions. Under full-slip conditions, a multiplicity of fractures initiate
and propagate at even spacing [3]. Pre-existing aws in the immediate vicinity
of the already developed fracture are shielded by it and hence are inhibited
from growth [40,43]. If the applied loads to indenter are higher than the critical loads, then they would eectively negate the shielding eect at a certain
distance away from the previously developed fracture. Beyond this critical
spacing, a pre-existing aw is not inhibited from growth. Thus, the spacing
of fractures is determined by the trade-o between the supercritical applied
loading and the shielding.
The contribution of the two competing eects on the stress intensity factor of
a pre-existing aw (located at an arbitrary distance d away from a developed
fracture) at the trailing end of a sliding indenter can be obtained by superposition. The rst component is due to the indenter loading which is independent
of the existing fractures and the same as given by equation (21).
The pre-existing aw will be modeled as a penny-shaped vertical crack at a
distance d from the already developed fracture. Stress-intensity factor of this
crack due to indentation tractions can be evaluated from equation (21). For a
xed normal load and contact radius, the horizontal stress at the surface under
Hertzian pressure varies inversely with (1 + d=a)2 [24]. Therefore the stressintensity factor at the tip of the developed fracture ( KIv ) can be estimated
by:
KIv (d) =
KIv (d = 0)
(1 + d=a)2
21
=
KIc
(1 + d=a)2
(28)
For d = 0, the fracture is at the trailing edge and its stress-intensity factor
must have been the same as the fracture toughness. The variation of horizontal
stress due to tangential tractions is more complex [24], but it also drops rapidly
with d and may also be approximated by variation 1=(1 + d=a)2 similar to the
Boussinesq stress eld for a point-load on a half-space.
For the shielding component, we need to estimate the perturbation of the stress
eld around the pre-existing aw due to the developed fracture. The dilation
of the latter induces a compressive stress which tends to inhibit the extensive
propagation of the former. If the pre-existing aw size is much smaller than
the fracture radius, then the horizontal component of this compressive stress
at the location of aw is proportional to the fracture stress intensity factor
KIv (d) (e.g. [44]):
xx (d) =
s
KIv
(d) 8d
(
2
? (1 ? 2 2)
(1 + )
1
1 + 2
? sin?1 p
)
(29)
where is dened as the ratio of d to the radius of the previously developed
fracture. The resultant stress-intensity factor KIf is therefore given by the sum
KIf (d) = KIf1 +
2xx (d) pc
f
(30)
where the rst term KIf1 is the stress-intensity factor of the aw cf in absence
of the fracture (given by equation (21) above) and the second term corresponds
to the shielding eect from the compressive stress given by equation (29). The
spacing of fractures can be determined by equating the above equation with
the fracture toughness.
If the applied loads are the same as the critical ones, then the negative shielding
eects can not be balanced and only one fracture will form. However, if the
indentation load is supercritical, then its eect will be dominant beyond a
certain distance since the shielding eect decays rapidly and is negligible at
distances larger than twice the fracture radius. Fig. 12 shows the variation of
stress-intensity factor with distance d when the applied load is 1.5 times the
critical load and friction coecient is 0.5. Flaw stress intensity factor increases
gradually with d and exceeds fracture toughness at a critical distance, beyond
which the aw extends to attain a nal length shown in Fig. 12. For very large
spacing, the nal length asymptotically approaches the size of the previously
developed fracture size. However, the aw which would grow at the critical
spacing has a somewhat smaller nal length. Consequently one expects that
the shielding eect of this second fracture would be less and therefore the
third fracture will extend to a longer length. This is consistent with Graham's
[45] observation that the size of fracture varies and that fracture spacing is
22
0.025
1.1
0.02
Fracture toughness
1
0.015
0.9
0.01
0.8
0.005
0.7
Final crack-size
Stress-intensity factor
1.2
0
0
0.5
1
1.5
Spacing relative to contact radius
2
0.8
0.100
0.090
0.6
0.080
0.4
0.070
0.2
0.060
0.0
Final crack-length
Spacing relative to contact radius
Fig. 12. Stress-intensity factor of a aw as a function of spacing with the existing
fracture. KI increases with distance and exceed fracture toughness at a critical
distance which would determine the spacing of fractures under a sliding indenter.
Final crack-length at large spacing is the same as the size of existing fracture,
although at the critical spacing its size is somewhat less due to the shielding eects.
0.050
1.0
1.2
1.4
Spacing (f=0.3)
Crack size
(f=0.3)/ Critical load
Normal
1.6
Spacing (f=0.5)
Crack size (f=0.5)
Fig. 13. Variation of spacing between fractures under a sliding indenter and the size
of fractures with applied load and friction coecient. Loads decrease the spacing,
whereas the friction increase it. Final crack-size is less for higher friction for the
same ratio of applied load with critical load primarily due to the reduced critical
load. Crack-size increases with the loads.
dierent with depth.
In Fig. 13, the critical spacing (solid curves) and corresponding nal crack
length (dashed curves) are plotted as functions of the normal load for two
dierent coecients of friction. The critical spacing decreases with increasing
normal load, and is higher for larger friction coecient for the same ratio of
applied loads to critical loads, similar to Bower and Fleck's [43] results for the
2-dimensional case. Final fracture sizes are smaller for higher friction coecients due to the reduced critical loads. Shielding eects of smaller fractures
would be less and hence the spacing increases with friction coecient. If the
spacing is small with respect to fracture size, then the nearest neighbor inter23
action is not sucient and shielding eects of several already formed fractures
may have to be accounted for. This would increase the critical spacing distance from the values in Fig. 13. The surface density of cracks due to a sliding
indenter would be inversely proportional to the spacing and would increase
with loads and decrease with friction coecient.
4.2 Comparison with laboratory and eld observations
Our results can be compared with observations on experimental and natural
sliding surfaces. Several recent studies were performed to characterize quantitatively the topography and roughness of fault surfaces (e.g. [[46,47]). In the
laboratory samples, the asperities have radii in the range of 20 to 100 m.
With a normal stress in the range of 10-100 MPa, and friction coecient of
0.5, the contact patches would have radii of 2-20 m [46] and the fracture
array would occur with periodic spacing of 1 to 20 m. The spacing being less
than a typical grain-size, most of the cracks will be intragranular similar to
the observations of Hundley-Go and Moody [48] for orthoquartzite with an
average grain-size of 150 m. In the eld, asperity radii span several orders of
magnitude and vary from few ms to several meters [49,47]. If fracture spacing
is similar to grain-size, then near-by fractures may coalesce at grain boundaries to generate a wear particle. For friction coecient of 0.5 and fracture
spacing equal to the grain diameter of 200 m, an asperity with a radius of
around 20 cm is required.
Our analysis suggests that if the stress level is suciently high, then a periodic array of fractures will be developed at the wake of a sliding asperity.
Furthermore, the spacing between successive fractures is so close that one expects the cumulative fracturing process to be very eective in generating the
wear grooves which are commonly observed in laboratory and eld samples
[17,18]. The width of wear groove and fracture density within it give quantitative information about the loads carried by an asperity and about its size.
The knowledge of contact loads may also be useful in estimating the real area
of contact. Total maximum distance that an asperity will slide is the amount
of slip and hence the length of wear groove cannot be larger than the slip.
The interaction of repeated slip events may hinder the prediction of slip based
on the groove-length measurements. The density of wear grooves will increase
with normal stress since more number of contact forces will have exceeded
the critical load. Engelder [17] reported that the number of grooves per slip
event increased by 3 orders of magnitude for an order of magnitude increase
in conning pressure.
It is also clear that since the sliding velocity does not enter the analysis,
groove will form independent of whether the slip is seismic or not, unless
24
the frictional stability behavior is sensitively dependent on the stress level.
Wear grooves were observed primarily in Engelder's [17] samples with stickslip behavior. Since his laboratory data showed frictional instability only under
elevated normal stresses, indentation stress levels at many of the asperities
are probably above the critical load for the development of Hertzian type of
fracturing. For further development of the theoretical analysis, it is desirable
to have simultaneous measurements of asperity statistics of the sliding surface
as well as the wear groove geometry and statistics, which will place important
constraints on the fracture mechanics parameters.
5 SUMMARY AND DISCUSSION
The present study identied the complete elastic stress elds in a half-space
due to an arbitrarily loaded blunt indenter. The tangential contact force and its
loading path alters the elds considerably and determine failure of the contact
in both brittle and plastic modes. The analytical results can be used to develop
the rst-order estimate for failure initiation loads, extent of plastic zone, and
size of fractures. Although such estimates disregard complex interaction of
cracking, plastic deformations, and the contact tractions, they provide a basis
for identifying qualitative role of relevant parameters. Partial-slip conditions
at a contact are more ecient in inducing failure or equivalently it is easy to
fail rough surfaces. The inelastic deformations underneath the contact may
continue even during the unloading of tangential force.
We evaluated the stress concentration factors for the existing aws located at
the site where tensile stresses are maximum and developed the condition for
fracture initiation and propagation for ideally brittle materials. Although the
fracture initiation may be unstable, its further propagation is stable. The tangential force which is just 10% of the normal force reduces the initiation load
by an order of magnitude. The model shows the decreasing critical loads with
the aw size and predicts the experimentally observed nonlinear dependency
on the indenter radius. The spacing of fractures under a sliding indenter were
estimated with model and the results were consistent with the other numerical
study and with the eld and experimental observations of wear grooves.
The fundamental understanding of contact deformations and failure is relevant
to many rock engineering and geological problems. The problem of fracturing
at grain contacts is one in which the plastic deformations may be negligible
and brittle rheology is truly applicable. It is found that the model based on
the Hertzian fracturing explains well the role of grain radius [10], cementation,
and global shear stresses in comminutive processes [21]. The present analysis
explains the initiation and stable growth of Hertzian cracks during hard-rock
indentation [6] and may allow us to obtain indirect measure of fracture tough25
ness and surface energy.
6 CONCLUSIONS
The analytical stress elds within spherical bodies subjected to arbitrary normal and tangential loads were obtained through the suitable superposition of
full-slip solution. The partial-slip situation increases both tensile and shear
stresses even though the contact is applied the same forces and shows the
importance of the friction coecient at the contact. During unloading of the
tangential force, the location of maximum tensile stress moves away from the
trailing edge towards the center and may be responsible for surface damage
during cyclic loading. The critical fracture load required for the development
of Hertzian crack system is reduced signicantly by the application of tangential load. The load is further reduced if the condition at the contact is of
partial-slip or if the contact is unloading. The analytical stress elds with a
simple estimate of shielding interaction between adjacent fractures were used
to predict spacing between the fractures under a sliding indenter. The spacing,
which is the inverse of surface fracture density, reduces with applied loads and
also with friction coecient.
Acknowledgement
This research was partially supported by the Oce of Basic Energy Sciences,
Department of Energy, under grant DEFG0294ER14455. We thank two anonymous reviewers for several useful comments.
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